441 research outputs found
Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs
In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric
Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field
, any infinite sequence of (skew) symmetric matrices
over of bounded -rank-width has a pair , such
that is isomorphic to a principal submatrix of a principal pivot
transform of . We generalise this result to -symmetric matrices
introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs,
arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of
-symmetric matrices. As a by-product, we obtain that for every infinite
sequence of directed graphs of bounded rank-width there exist a
pair such that is a pivot-minor of . Another consequence is
that non-singular principal submatrices of a -symmetric matrix form a
delta-matroid. We extend in this way the notion of representability of
delta-matroids by Bouchet.Comment: 35 pages. Revised version with a section for directed graph
Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices
We prove that every infinite sequence of skew-symmetric or symmetric matrices
M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such
that M_i is isomorphic to a principal submatrix of the Schur complement of a
nonsingular principal submatrix in M_j, if those matrices have bounded
rank-width. This generalizes three theorems on well-quasi-ordering of graphs or
matroids admitting good tree-like decompositions; (1) Robertson and Seymour's
theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's
theorem for matroids representable over a fixed finite field having bounded
branch-width, and (3) Oum's theorem for graphs of bounded rank-width with
respect to pivot-minors.Comment: 43 page
When Can Matrix Query Languages Discern Matrices?
We investigate when two graphs, represented by their adjacency matrices, can be distinguished by means of sentences formed in MATLANG, a matrix query language which supports a number of elementary linear algebra operators. When undirected graphs are concerned, and hence the adjacency matrices are real and symmetric, precise characterisations are in place when two graphs (i.e., their adjacency matrices) can be distinguished. Turning to directed graphs, one has to deal with asymmetric adjacency matrices. This complicates matters. Indeed, it requires to understand the more general problem of when two arbitrary matrices can be distinguished in MATLANG. We provide characterisations of the distinguishing power of MATLANG on real and complex matrices, and on adjacency matrices of directed graphs in particular. The proof techniques are a combination of insights from the symmetric matrix case and results from linear algebra and linear control theory
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model
Starting from the Verma module of U_q sl(2) we consider the evaluation module
for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an
associated integrable statistical mechanics model on a square lattice defined
in terms of vertex configurations. Its transfer matrix is the generating
function for noncommutative complete symmetric polynomials in the generators of
the affine plactic algebra, an extension of the finite plactic algebra first
discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative
elementary symmetric polynomials were recently shown to be generated by the
transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin
and Kitanine. Here we establish that both generating functions satisfy Baxter's
TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions
of the Yang-Baxter equation. The TQ-equation amounts to the well-known
Jacobi-Trudy formula leading naturally to the definition of noncommutative
Schur polynomials. The latter can be employed to define a ring which has
applications in conformal field theory and enumerative geometry: it is
isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure
constants are the dimensions of spaces of generalized theta-functions over the
Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe
The su(n) WZNW fusion ring as integrable model: a new algorithm to compute fusion coefficients
This is a proceedings article reviewing a recent combinatorial construction
of the su(n) WZNW fusion ring by C. Stroppel and the author. It contains one
novel aspect: the explicit derivation of an algorithm for the computation of
fusion coefficients different from the Kac-Walton formula. The discussion is
presented from the point of view of a vertex model in statistical mechanics
whose partition function generates the fusion coefficients. The statistical
model can be shown to be integrable by linking its transfer matrix to a
particular solution of the Yang-Baxter equation. This transfer matrix can be
identified with the generating function of an (infinite) set of polynomials in
a noncommutative alphabet: the generators of the local affine plactic algebra.
The latter is a generalisation of the plactic algebra occurring in the context
of the Robinson-Schensted correspondence. One can define analogues of Schur
polynomials in this noncommutative alphabet which become identical to the
fusion matrices when represented as endomorphisms over the state space of the
integrable model. Crucial is the construction of an eigenbasis, the Bethe
vectors, which are the idempotents of the fusion algebra.Comment: 33 pages, 8 figures; published in conference proceedings (Kokyuroku
Bessatsu) of the workshop "Infinite Analysis 10, Developments in Quantum
Integrable Systems", Research Institute for Mathematical Sciences, Kyoto,
June 14-16, 2010; v2: some typos in Section 4 fixe
Polygon gluing and commuting bosonic operators
We construct two series of commuting Hamiltonians parametrized by a constant
matrix. The first series was my guess and the second one was know, and in our
approach follows from the first series. For the proof we use familier facts
known from our previous consideration of the links between random matrices and
Hurwitz numbers, however the text is self-consistent.Comment: 9 pages, 1 figur
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